Statistics for Environmental Engineers

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The Working-Hotelling confidence band for the entire line is computed from:


b0 + b1 x ± l2F2,v,aS2(~ + (X ^ ■


V    vn    £(xi — x)


in which


2 = Е(у,~у)2 S    n — 2


Note that this confidence band is not the same as the confidence interval for у/ predicted from a future value of x, a case that was described in Chapter 34. This confidence region can be narrowed somewhat by limiting the range of the abscissa variable (Wynn and Bloomfield, 1971; Hunter, 1981).


Leiberman et al. (1967) proposed a practical approach to using the Working-Hotelling confidence region for the calibration line to obtain a confidence interval for the predicted concentration. A 100P% confidence interval for the true response (n) based upon a future observation у is computed using:


.    . vs


у ± N 7″


v,a/2


where у is the measured instrument response and zp is the standard normal deviate for percentage P. The quantity under the square root is an estimate of the upper (1 — a )100% confidence limit on the standard deviation of the instrument response (у) where s is the variance of replicate measurements of у.


Xv,a/2 is the lower a/2 percentile point of the X distribution, where v = n — 2 degrees of freedom associated with the variance estimate (s2).


This interval for у and the Working-Hotelling confidence band for the calibration line are used to estimate a confidence interval about x for the true concentration |. This typically is done graphically. Hunter (1981) provides a full derivation and explanation of the procedure.

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